Pierpont prime

A Pierpont prime with is of the form 2^u+1, and is therefore a Fermat prime (unless ). If is positive then must also be positive (because 3^v+1 would be an even number greater than 2 and therefore not prime), and therefore the non-Fermat Pierpont primes all have the form , when is a positive integer (except for 2, when ). Empirically, the Pierpont primes do not seem to be particularly rare or sparsely distributed; there are 42 Pierpont primes less than 106, 65 less than 109, 157 less than 1020, and 795 less than 10100. There are few restrictions from algebraic factorisations on the Pierpont primes, so there are no requirements like the Mersenne prime condition that the exponent must be prime. Thus, it is expected that among -digit numbers of the correct form 2^u\cdot3^v+1, the fraction of these that are prime should be proportional to , a similar proportion as the proportion of prime numbers among all -digit numbers. As there are \Theta(n^{2}) numbers of the correct form in this range, there should be \Theta(n) Pierpont primes. Andrew M. Gleason made this reasoning explicit, conjecturing there are infinitely many Pierpont primes, and more specifically that there should be approximately Pierpont primes up to . According to Gleason's conjecture there are \Theta(\log N) Pierpont primes smaller than N, as opposed to the smaller conjectural number O(\log \log N) of Mersenne primes in that range. ==Primality testing==

When 2^u > 3^v, 2^u\cdot 3^v + 1 is a Proth number and thus its primality can be tested by Proth's theorem. On the other hand, when 2^u alternative primality tests for M=2^u\cdot 3^v + 1 are possible based on the factorization of M-1 as a small even number multiplied by a large power of 3. == Pierpont primes found as factors of Fermat numbers ==

As part of the ongoing worldwide search for factors of Fermat numbers, some Pierpont primes have been announced as factors. The following table gives values of m, k, and n such that {{Block indent|left=1.6|2^{2^m} + 1 is divisible by 3^{k} \cdot 2^{n} + 1.}} The left-hand side is a Fermat number; the right-hand side is a Pierpont prime. , the largest known Pierpont prime is 81 × 220498148 + 1 (6,170,560 decimal digits), whose primality was discovered in June 2023. ==Polygon construction==

In the mathematics of paper folding, the Huzita axioms define six of the seven types of fold possible. It has been shown that these folds are sufficient to allow the construction of the points that solve any cubic equation. It follows that they allow any regular polygon of sides to be formed, as long as and is of the form , where is a product of distinct Pierpont primes. This is the same class of regular polygons as those that can be constructed with a compass, straightedge, and angle trisector. As Gleason later showed, these numbers are exactly the ones of the form given above. All other regular with can be constructed with compass, straightedge and trisector. == Generalization ==

A Pierpont prime of the second kind is a prime number of the form 2u3v − 1. These numbers are The largest known primes of this type are Mersenne primes; currently the largest known is 2^{136279841}-1 (41,024,320 decimal digits). The largest known Pierpont prime of the second kind that is not a Mersenne prime is 3\cdot 2^{22103376}-1, found by PrimeGrid. A generalized Pierpont prime is a prime of the form p_1^{n_1} \!\cdot p_2^{n_2} \!\cdot p_3^{n_3} \!\cdot \ldots \cdot p_k^{n_k} + 1 with k fixed primes p1 2 3 k. A generalized Pierpont prime of the second kind is a prime of the form p_1^{n_1} \!\cdot p_2^{n_2} \!\cdot p_3^{n_3} \!\cdot \ldots \cdot p_k^{n_k} - 1 with k fixed primes p1 2 3 k. Since all primes greater than 2 are odd, in both kinds p1 must be 2. The sequences of such primes in the OEIS are: ==See also==